A lower bound for the Lindelöf function associated to generalized integers

In this paper we study generalized prime systems for which the integer counting function NP(x)NP(x) is asymptotically well-behaved, in the sense that NP(x)=ρx+O(xβ)NP(x)=ρx+O(xβ), where ρ   is a positive constant and β<12. For such systems, the associated zeta function ζP(s)ζP(s) has finite order for σ=Rs>β, and the Lindelöf function μP(σ)μP(σ) may be defined. We prove that for all such systems, μP(σ)⩾μ0(σ)μP(σ)⩾μ0(σ) for σ>βσ>β, whereμ0(σ)={12−σif σ<12,0if σ⩾12.